require "algebra"
Z13 = ResidueClassRing(Integer, 13)

a, b, c, d, e, f, g = Z13
p [e + c, e - c, e * c, e * 2001, 3 + c, 1/c, 1/c * c, d / d, b * 1 / b]
  #=> [6, 2, 8, 9, 5, 7, 1, 1, 1]
p( (1...13).collect{|i|  Z13[i]**12} )
  #=> [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]